1. Perfect and Related Codes

2. OUTLINE [1] Some bounds for codes [2] Perfect codes [3] Hamming codes
[4] Extended codes
[5] The extended Golay code
[6] Decoding the extended Golay code
[7] The Golay code
[8] Reed-Muller codes
[9] Fast decoding RM(1,m)

3. Perfect and Related Codes
[1] Some bounds for codes
1. The number of word of length n , weight t
2. Theorem 3.1.1

4. Perfect and Related Codes
3. Theorem Hamming bound(upper bound)
C: a code of length n, distance d = 2t+1 or 2t+2
Eg Give an upper bound of the size of a linear code C of length n=6 and distance d=3
So but the size of a linear code C must be a power of 2 so

5. Perfect and Related Codes
4. Theorem Singleton bound(upper bound)
For any (n, k, d) linear codes, d-1≦ n-k
(i.e. k ≦ n-d+1 or |C| ≦2n-d+1 )
<pf> the parity check matrix H of an (n,k,d) linear code is an n by
n-k matrix such that every d-1 rows of H are independent.
Since the rows have length n-k, we can never have more than
n-k independent row vectors. Hence d-1≦ n-k.
5. Theorem 3.1.8
For a (n, k, d) linear code C, the following are equivalent:
d = n-k+1
Every n-k rows of parity check matrix are linearly independent
Every k columns of the generator matrix are linearly independent
C is Maximum Distance Separable(MDS) (definition: if d=n-k+1)

6. Perfect and Related Codes
6. Theorem Gilbert-Varshamov condition
There exists a linear code of length n, dimension k and distance d if
(<pf> design a parity check matrix under this condition. See Ex3.1.22)
7. Corollary Gilbert-Varshamov bound(lower bound)
If n≠1 and d ≠1, there exists a linear code C of length n and distance at least d with
(<pf> choose k such that
then |C| = 2k = )

7. Perfect and Related Codes
Eg Does there exist a linear code of length n=9, dimension k=2, and distance d=5? Yes, because
Eg What is a lower and an upper bound on the size or the dimension, k, of a linear code with n=9 and d=5?
G-V lower bound: |C| ≧ but |C| is a power of 2 so |C| ≧ 4
Hamming upper bound: |C| ≦
but |C| is a power of 2 so |C| ≦ 8

8. Perfect and Related Codes
Eg Does there exists a (15, 7, 5) linear code?
Check G-V condition
G-V condition does not hold, so G-V bound does not tell us
Whether or not such a code exists.
But actually such a code does exist. (See BCH code later)

9. Perfect and Related Codes
[2]. Perfect Codes
1. Definition:
A code C of length n and odd distance d = 2t+1 is called perfect code if
2. Theorem 3.2.8
If C is perfect code of length code of length n and distance d = 2t+1, then C will correct all error pattern of weight less than or equal t

10. Perfect and Related Codes
[3]. Hamming Codes
1. Definition: Hamming code of length 2r-1
A code of length n = 2r-1, r ≧2, having parity check matrix H whose rows of all nonzero vectors of length r
Eg the Hamming code of length 7(r=3)

11. Perfect and Related Codes
H contains r rows of weight one, so its r columns are linearly independent.
Thus a Hamming code has dimension k=2r-1-r and contains 2k codewords.
It is a perfect single error-correcting code (d=3)
1. Any two rows of H are lin. indep so d ≧3
2. 100…0, 010…0, and 110…0 are lin. dep. so d ≦3 and thus d=3
3. Attains the Hamming bound (t=1)

12. Perfect and Related Codes
[4]Extended Codes
1. Definition:
A code C* of length n+1 obtained from C of length n
Construction:
Let G: kxn and choose G*=[G, b] : kx(n+1) where the last col b of G* is appended so that each row of G* has even weight.
Let H: nx(n-k) then H* = where j is the nx1 col of all ones.
Why?
where Gj+b=0

13. Perfect and Related Codes
Eg 3.4.1
C: length 5
C*: length 6

14. Perfect and Related Codes
[5]. The extended Golay code(C24)
1. Definition
the linear code C24 with generator matrix G= [I, B]
I: 12x12 identity matrix B:

15. Perfect and Related Codes
2. Important facts of the extended Golary Code C24
Length n =24, dimension k= 12 , 212=4096 codewords
Parity check matrix
Another parity check matrix
Another generator matrix [B, I]
C24 self-dual;
The distance of C is 8
C24 is a three-error-correcting code

16. Perfect and Related Codes
[6]. Decoding the extended Golay code
1. Algorithm IMLD for Code C24
w: received word
1) s = wH
2) if wt(s)≦3 then u = [ s,0 ](error pattern)
3) if wt(s+bi) ≦2 for some row bi of B then u=[s+bi, ei]
4) compute the second syndrome sB
5) if wt(sB)≦3 then u=[0, sB]
6) if wt(sB+bi)≦2 for some row bi of B then u=[ei, sB+bi]
7) if u is not yet determined then request retransmission

17. Perfect and Related Codes
2. Eg
decode w = ,

18. Perfect and Related Codes
[7]. The Golay code
1. C23
Removing a digit from every word in C24
: 12 x 11 matrix by deleting the last column of B
G: 12x23 matrix, G=[I12, ]
Length n=23, dimension k=12, 212=4096 words, distance d=7
Perfect code, three-error-correcting code
Algorithm (decoding)
Form w0 or w1, which has odd weight
Decode wi(I is 0 or 1) using Algorithm to a codeword c in C24
Removing the last digit from c

19. Perfect and Related Codes
[8]. Reed-Muller codes
1. r-th order, length 2m, 0≦r≦m, RM(r, m)
RM(0, m)={00…0, 11…1}, RM(m, m)=
RM(r, m) =
2. Eg 3.8.1

20. Perfect and Related Codes
3. Generator matrix G of RM(r, m)
4. Eg Find G(1,3)

21. Perfect and Related Codes
5. The properties of RM(r,m)

22. Perfect and Related Codes
[9]. Fast decoding for RM(1,m)
1. The Kronecker product of matrices
A x B = [aijB]
Eg 3.9.1
Definition:

23. Perfect and Related Codes
2. Algorithm decoding the RM(1,m) code
codeword v = presumed message m x G(1,m)

24. Perfect and Related Codes
Eg m=3, w=
codeword